1
Social similarity favors cooperation: the
distributed content replication case
Eva Jaho, Merkourios Karaliopoulos, and Ioannis Stavrakakis
Abstract—This paper explores how the degree of similarity within a social group can dictate the behavior of the individual nodes,
so as to best trade-off the individual with the social benefit. More specifically, we investigate the impact of social similarity on the
effectiveness of content placement and dissemination. We consider three schemes that represent well the spectrum of behaviorshaped content storage strategies: the selfish, the self-aware cooperative, and the optimally altruistic ones. Our study shows that when
the social group is tight (high degree of similarity), the optimally altruistic behavior yields the best performance for both the entire group
(by definition) and the individual nodes (contrary to typical expectations). When the group is made up of members with almost no
similarity, altruism or cooperation cannot bring much benefit to either the group or the individuals and thus, selfish behavior emerges as
the preferable choice due to its simplicity. Notably, from a theoretical point of view, our ”similarity favors cooperation” argument is inline
with sociological interpretations of human altruistic behavior. On a more practical note, the self-aware cooperative behavior could be
adopted as an easy to implement distributed alternative to the optimally altruistic one; it has close to the optimal performance for tight
social groups and the additional advantage of not allowing mistreatment of any node, i.e., its induced content retrieval cost is always
smaller than the cost of the selfish strategy.
Index Terms—Content Replication, Cooperation, Similarity, Social Groups.
✦
1
I NTRODUCTION
Networks today can be highly personalized, in the sense
that their structure and usage are shaped by the personal
interests, or behavior in general, of the participating
nodes. Nodes in such networks – referred to as social
networks – are typically well connected, develop reciprocal trust relations, and share some attributes, such as
content interests and locality. Groups of such nodes are
called social groups [27].
In this paper we consider a group of networked
nodes with common preferences for content –or, more
generally, data objects such as files and software. The
node-members of this social group can store content in
their limited local storage and retrieve it when desired at
a minimum cost. When a desired object is not stored locally, nodes may either fetch it from the cache of another
node-member of the group at low-medium cost or, if the
object is not available within the group, from a node
outside the group at a higher cost. The low-medium
cost associated with fetching an object from within the
group may reflect low actual or virtual price such as
lower access delay due to locality or high connectivity, or
higher level of trust and reliability. We assume that nodes
have established trust relationships in order to belong to
the same social group, and have access to each others’
caches. This may be realized through various schemes
(e.g., see [11], [26], [28]).
• The authors are with the Department of Informatics and Telecommunications, National & Kapodistrian University of Athens, Ilissia, 157 84
Athens, Greece.
E-mail: {ejaho, mkaralio, ioannis}@di.uoa.gr
Since the local storage is considered to be limited
when compared with the plethora of possibly interesting
content, an inherently selfish node would opt for storing
locally objects of highest interest to it. Our past work in
[19] has shown that, in a distributed group with three
levels of content access cost as considered here, this is not
the best content placement strategy for a node. Instead,
a cooperative content placement strategy has been devised on the basis of game-theoretic arguments, whereby
nodes exchange information about their content placements and synchronize in adjusting them. This way they
avoid storing objects that could be fetched efficiently
from other nodes so that the resulting cost for each and
every node is at most (and typically much lower than)
that induced under the selfish strategy (mistreatmentfree property). In the present paper, we refer to this
placement strategy as the self-aware cooperative strategy,
to emphasize its cooperative nature and mistreatmentfree property.
Mistreatment-free strategies are key to the sustainability of such distributed selfish groups, as they motivate
the participation in the group and sharing of objects.
The social benefit induced by the self-aware cooperative
strategy, i.e., the aggregate benefit of all group nodes, is
not in general optimal. The content placement strategy
that maximizes the social benefit will be referred to
hereafter as the optimally altruistic strategy; it can be
derived by solving an optimization problem, as done, for
instance, in [20]. The practical implementation of the optimally altruistic strategy requires the exchange of richer
information among the nodes in the group (complete local demand distributions); whereas, under the self-aware
cooperative strategy, nodes need to exchange much less
2
information (indices of locally stored content). Finally,
although the optimally altruistic strategy maximizes the
aggregate benefit, some of the nodes may end up with
high gains and others being mistreated.
Focus of this paper: In view of the above it is evident
that a node participating in a distributed group can face
a dilemma as to which strategy to follow.
• The selfish strategy requires no interaction with and
guarantees no mistreatment by the other nodes; on
the other hand, both the node itself and the group
could benefit more by following another strategy.
• The self-aware cooperative strategy outperforms the
selfish strategy both at the individual node and
group levels, while it ensures no mistreatment of
individual nodes; on the other hand, it does not
maximize the group benefit, while it introduces
complexity that increases with the group size and
may outweigh the benefits for individual nodes.
• The optimally altruistic strategy yields the maximum
possible benefit for the entire group. On the other
hand, it can mistreat certain nodes (with the risk of
inciting them to leave the group) and requires heavier interaction with the other nodes of the group
(increasing complexity); these interactions could be
lighter under a centralized derivation of the optimal
placement.
In this paper the characteristics of the social group are
exploited to address the above dilemma. To this end, we
follow an innovative approach to the characterization of
the nodes’ similarity within a social group and introduce
a group tightness metric, which explicitly accounts for
the level of similarity in their content preferences. Our
work highlights the impact of the metric on the induced
social and individual node benefits under the three aforementioned content placement strategies, which reflect
general patterns of social behavior. It draws important
insights as to how rewarding such behaviors can be
under given levels of social group tightness.
This study has applications to social networks featuring interactions between computer devices with limited
memory resources. These are typically encountered in
mobile opportunistic networks that are additionally “socially aware”, meaning that either the nodes or their
human users are aware of the formation of social groups
and the potential benefits from participation in such
a group. The underlying assumption is that there are
multiple groups, and we focus on the behavior regarding the exchange of information objects between nodes
inside a single group. Studying content access patterns,
especially between nodes in a social group is quite
important to assess the viability of various networking
paradigms, e.g., the opportunistic wireless networking
and some P2P systems.
The remainder of this paper is organized as follows.
In Section 2 we present the three content placement
strategies and their main properties. We introduce the
tightness metric, a novel metric for capturing the similarity of content preferences within the group based
on the Kullback-Leibler divergence between preference
distributions, and discuss its appropriateness in Section 3. The evaluation methodology and scenarios are
presented in Section 4. In Section 5 we analyze the
selfish and self-aware cooperative placement strategies
and derive sufficient conditions under which the latter
yields the same placements with the selfish and the
optimal strategies. The content placement strategies are
numerically compared under different tightness values
in Section 6. We also assess the convergence time of
the cooperative algorithm in the more realistic scenario
that nodes gradually learn the content preferences of
the users behind them and adapt their placements accordingly. Our conclusions about the cooperation gains
achievable under different tightness values are further
validated in Section 7, where the content preferences of
users are extracted by crawling data from an online social bookmarking service. We contrast our contributions
against related work in Section 8 and summarize the
major conclusions of the paper in Section 9, drawing
parallels to outcomes of sociological studies of human
behavior.
2
C ONTENT P LACEMENT S TRATEGIES
AND
RELATED TRADEOFFS
Let N = {1, 2, ..., N } denote the set of the nodes in a
social group and let M = {1, 2, ..., M } denote the set of
n
objects (or items) these nodes are interested in. Let Fm
denote the preference probabilities of node n, for object
n
n
}; Fm
can be viewed as
m, and let F n = {F1n , F2n , ..., FM
the normalized rate of requests for object m by node n.
Let Pn denote the placement at node n, defined to
be the set of objects stored locally at that node with
storage capacity Cn . All objects are considered to be of
the same size. Throughout the paper, we are interested
in placements of cardinality |Pn | = Cn since, under
any placement strategy, the node has always motivation
to fully utilize its buffer space. More formally, for any
placement Pn with |Pn | < Cn , we can always find
at least one other placement Pn′ with Pn ⊆ Pn′ and
|Pn′ | = Cn , which (weakly) dominates Pn in terms of cost.
Let P = {P1 , P2 , . . . , PN } denote the global placement
for the social group and P−n = P \ Pn denote the set of
placements for all group nodes but node n.
Let tl , tr and ts denote the cost for accessing an object
from the node’s local memory, from another remote node
within the social group and from nodes in another social
group, respectively; tl < tr < ts . These costs are assumed
to be the same for all nodes in order to simplify the
analysis. The most problematic assumption is that the
cost of accessing an object from any group node is the
same irrespective of the involved nodes, i.e., tl < tij
r =
tr < ts , ∀i, j ∈ N . However it may be reasonable when
the group draws on locality or related types of social
context.
If the nodes of the group are in proximity the access cost could either represent the additional latency
3
incurred when fetching content or the bandwidth consumed when retrieving content, depending on the scenario of interest. Otherwise, the low-medium cost associated with fetching an object from within the group may
reflect low price due to high level of trust.
Given an object placement P, the mean access cost incurred to node n per unit time for accessing its requested
objects is given by:
X
X
X
n
n
n
Cn (P) =
Fm
ts .
Fm
tl +
Fm
tr +
(2.1)
m∈P
/ n,
m∈P
/ n,
m∈P
n
m∈P−n
m∈P
/ −n
The first, second and third addends on the right hand
side correspond to the mean cost for accessing objects
locally, from other nodes of the group and from external
sites if not found within the group, respectively. The
object placement strategies considered in this paper are
described next.
Optimally altruistic strategy: the objects are stored in
such a way that the total access cost for all nodes in the
PN
social group is minimized (i.e., , minimize n=1 Cn (P)).
This problem can be transformed into a 0-1 integer
programming(problem.
1, if m ∈ Pn ;
n
Let Xm
=
and
0, otherwise
(
1, if m ∈
/ Pn and m ∈ P−n ;
n
Ym =
0, otherwise.
The objective is to minimize the total access cost:
M
N X
X
n n
n
[Xm
Fm tl + Ymn Fm
tr +
n=1 m=1
N
Y
j
n
(1 − Xm
)Fm
ts ], (2.2)
j=1
where
n
Ymn = (1 − Xm
)(1 −
N
Y
j
(1 − Xm
)).
(2.3)
j=1
j6=n
This is a special case of a quadratic programming
problem with zero diagonal elements, whose solution
is very difficult [22]. It was shown in [20] that this
quadratic problem reduces to a 0-1 integer linear minimization problem (ILP) with objective function
f (X) =
N
+1 X
M
X
n
n
zm
Xm
,
(2.4)
n=1 m=1
PN +1
n
n=1 Xm
≥ 1, 1 ≤ m ≤ M and
subject to
PM
n
1 ≤ n ≤ N . In (2.4), the terms
m=1 Xm ≤ Cn ,
n
Xm
are as above, the additional virtual node N + 1
represents the ensemble of nodes in other social
n
groups,
the terms zm
, n ∈ N , are defined as
( and
n
Fm
(tr − tl ),
for 1 ≤ n ≤ N ;
n
zm
=
PN
j
j=1 Fm (tr − ts ), for n = N + 1,
In this ILP formulation, there is effectively an implicit
reference placement, whereby all nodes can access all
objects from the caches of group nodes and aggregate
P
PM
n
access cost N
n=1
m=1 Fm tr . The aim is then to derive
object placements that improve over this reference
n
placement. Hence, the terms zm
, n ∈ N , express the
incremental benefit resulting for each node when it
stores the object locally instead of retrieving it from
N +1
the group; whereas, zm
notes the loss all nodes incur
if the object is not stored anywhere in the group. The
equivalence of the quadratic maximization problem
(2.2) to the minimization ILP (2.4) is further explicated
in [14].
Selfish strategy: Under the selfish (or Greedy Local, as
referred to in [19]) strategy, the nodes store the objects
they prefer most. Each node n ranks the objects in
decreasing order of preference and selects to store the
first Cn ones.
Self-aware cooperative strategy: The placement strategy evolves in two rounds. First, each node stores its
Cn most preferable items (as with the selfish placement
strategy). Then, nodes take turns in adjusting their placements based on the placements of the other nodes in
the group. During this second round, each node has
the chance to replace some or all its items in order to
come up with the most cost-effective placement given
the placements of other nodes. Thus, a node may decide
to evict an object stored in some other node in the group
and insert a new one, if this reduces its access cost
according to (2.1). As the nodes amend their placements
sequentially, each replacement made by one node affects
both the access cost of nodes that have already made
their adjustments and the choices made by nodes that
follow.
In Table 1 we summarize results for the implementation cost (over all nodes) for the three strategies, considering the computational complexity cost. The details of the
derivation are provided in [14] and are also available in
[19]. The computational complexity refers to the cost for
all nodes to decide which objects to store locally.
Table 1: Computational cost
Strategy
Selfish
Self-aware cooperative
Optimally altruistic
Computational
O(N M logM )
O(N M logM )
NP-hard
2.1 Properties of the self-aware cooperative strategy
The self-aware cooperative strategy is a distributed localsearch algorithm for solving the distributed content
placement game [19]. This is an N-player noncooperative
game, whereby the players (nodes) behave as rational
selfish agents that aim to minimize their aggregate content access cost. Each node is calledto select its own pure
strategy(placement) Pn among CMn alternatives and the
aggregate placement P resulting from the combination
of the nodes’ choices presents each node with a payoff
Cn (P), given by (2.1).
The algorithm evolves in two rounds. During the
first round, the nodes move asynchronously without
4
sharing any information; whereas, in the second round,
the nodes play sequentially following a predefined order.
By the time the node playing in the k th position has to
determine its own placement, it knows everything that
the node who played before it knew, i.e., the placements
of the first k − 2 nodes, plus the placement of the
node in position k − 1. Hence, this second round of
the algorithm points to an N-player single-act dynamic
game and, irrespective of the particular order of play,
can be represented in ladder-nested extensive form [2]1 .
In [19], it has been shown that under the self-aware
cooperative placement strategy nodes may evict from
their caches only objects that are replicated elsewhere
in the group (property P1); nodes may insert in their
caches only objects that are not represented in the group
(property P2); and the placements do not give rise to node
mistreatment phenomena, i.e., for any node n, it holds
that CnC (P) ≤ CnS (P), where CnC (P) and CnS (P) denote
the mean access cost for node n under the self-aware
cooperative and selfish strategy, respectively (property
P3) (for a discussion of these properties when the withingroup access costs are not symmetrical, refer to [14]).
More importantly, it has been shown that, irrespective
of the order of play, the resulting global placement
after all nodes make their amendments in the second
round, is a Nash Equilibrium (NE) in that no node
has a reason to unilaterally change its own placement.
In other words, the self-aware cooperative placement
always yields placements that are NE in pure strategies.
What changes with the order of play is the resulting
placement P = {P1 , P2 , . . . , PN }, hence the individual
and social costs corresponding to the NE. In general,
the number of different NE ranges in [1, D], where D =
N
min(N !, M
); the first number reflects the possible
C
permutations in the order nodes play and the second
determines all the possible ways the M content objects
can be replicated in the caches of N nodes.
The question then becomes how do the placements
under the self-aware cooperative strategy compare with
the optimal and, secondarily, the selfish ones. From
a distributed algorithm viewpoint, the search is for
an (ideally, constant and tight) approximation ratio; in
game-theoretic terms, we are after the price-of-anarchy
(PoA) of the game, i.e., the ratio of the social cost under
the worst possible NE over the optimal cost [16]. We
elaborate on this later in Section 5.
3
G ROUP
TIGHTNESS METRIC
Each one of the three placement strategies presented
in Section 2 resolves differently the multiple tradeoff
among: a) the performance of individual nodes and of
the entire group; b) the possibility of individual nodes
1. The game has also connections with leader-follower (Stackelberg)
games [2]. However, the nodes do not play in Stackelberg mode, i.e.,
assuming a priori that each subsequent node will play best-response
to the future states of the game (node placements); they rather play
themselves best-response to the current state of the game.
being mistreated and the respective (lack of) incentives
for cooperation; c) the required communication overhead
and computational complexity for each strategy realization. The obvious question then for a node-member of
the social group is which strategy is the most “appropriate” to follow. In this section, we introduce a metric,
which we call tightness, for the similarity of interests
within the social group that can be of help in reaching
a conclusion.
The definition of tightness draws on the symmetrized
Kullback-Leibler (K-L) divergence [17], a well-known
measure of divergence between two distributions. The
Kullback-Leibler divergence of distribution Q from S is
defined as:
X
S(i)
.
S(i)log
DS,Q =
Q(i)
i
and its symmetrized counterpart is D(S||Q) = DS,Q +
DQ,S .
The average divergence of nodes’ preferences within
the group can then be written as:
P
i
j
(i,j) D(F ||F )
,
(3.1)
D̂F =
N (N − 1)/2
where the summation above is carried out over all
N (N − 1)/2 node pairs (i, j). Finally, we define tightness
T to be the inverse of D̂F :
1
T =
.
(3.2)
D̂F
We elaborate on computational aspects of the tightness
metric in [14].
3.1 Why tightness as a metric?
In principle, various measures of distributional similarity could quantify the similarity of content preferences
across the nodes of a group, such as the Spearman’s
rank correlation coefficient [24], Kolmogorov-Smirnov
distance [31], proportional similarity [30], and total variation distance [10]. Compared to them, the proposed
tightness metric has the following advantages:
Sensitivity to rank-preserving dissimilarity: Contrary to
metrics such as Spearman’s rank correlation coefficient,
tightness can capture dissimilarity of interests among
nodes that may rank the objects similarly, yet focus with
different intensity on the top-k content objects (see rankpreserving dissimilarity case in Section 4).
Account of full preferences’ profiles: Contrary to the
the Kolmogorov-Smirnov (K-S) distance metric, which
considers the supremum of the differences over all elements of a distribution, the K-L divergence accounts
for deviations across the whole distribution. Thus, the
proposed tightness metric captures more accurately the
overall distributional (dis)similarity.
Broader range of values: In contrast with the proportional similarity and total variation distance metrics [30],
which yield values in [0, 1], tightness values vary in
(0, +∞). Therefore it can resolve easier finer levels of
5
distributional divergence. In recent work [13], we have
shown how this property of the tightness metric can
benefit a different task, that of community detection, by
modulating its resolution.
Finally, we should note that when some element values of one of the distributions are zero while the corresponding elements of the other distribution are not (i.e.,
the request rate of a node for an object is zero), the K-L
distance value approaches infinity. In order to avoid such
problems, smoothing methods such as interpolation and
backing-off schemes can be used for providing reliable
probability estimates. These methods have been studied
in statistical language modelling in order to estimate the
distribution of natural language elements as accurately
as possible. In our case, non-zero request rates for objects
can be discounted with different discounting methods
(see [23]), whereas all other non-requested objects can
be given a minimal ǫ probability. In this paper, we
will consider that all nodes have probability mass (i.e.,
positive request rate) for all content objects, so that we
do not need to apply any smoothing method.
4
E VALUATION
METHODOLOGY AND SCENAR -
IOS
Tightness expresses the similarity of preferences among
the nodes of the social group and is always greater than
or equal to zero. In fact, T → ∞ when the group nodes
have very similar preferences and T → 0 when they have
very diverse preferences. As T is an average metric over
all node pairs of a social group, it is clear that a given
value of T may arise under different combinations of
node-level content preference distributions.
The content preferences of nodes are modeled by Zipf
distributions with variable shape parameter s, i.e., the
normalized interest of node n for its k th most interesting
PM
object is (1/k)s / l=1 1/ls . Zipf distributions combine
modeling simplicity with flexibility in that proper manipulation of their shape parameter s, gives rise to a
wide set of distributions ranging from uniform (s = 0) to
the highly skewed ones with power-law characteristics
(s >> 0)2 .
In order to draw more insightful conclusions in the
current study, we distinguish between the following
two broad patterns of dissimilarity in the preference
distributions.
4.1 Rank-preserving dissimilarity
The rank of the objects remains the same for all group
member nodes, i.e., the ith most popular object for all
nodes is the same, i ∈ [1, M ]. However, the mass of
the distributions concentrates more towards the highlyranked objects as the shape parameter s increases.
2. Zipf distributions have also been shown to be good models of content popularity both within and across different Internet Autonomous
Systems [12], and have been used widely in the literature in this
respect.
n
More specifically, the preferences Fm
for the object
m, m ∈ [1, M ], are drawn from Zipf distributions with
different exponent sn for each node. We let s1 = 0
for the first node (uniform interest distribution) and
sn = p(n − 1) for node n, n ∈ [2, N ], where p ∈ R
is the increment parameter. As shown in Table 2(a),
under the (object-)rank-reserving dissimilarity scenario,
tightness is a monotonically decreasing function of p. As
p increases, the content preference distributions of nodes
diverge more strongly, resulting in higher pairwise K-L
divergence values between any two node distributions.
4.2 Shape-preserving dissimilarity
The preference distributions are identical in shape for all
nodes, yet the ranking of a given object differs from node
to node, i.e., the k th , 1 ≤ k ≤ M , most popular object for
each node is different. The dissimilarity of nodes can be
more dramatic in this case and lower tightness values
are expected on average.
Contrary to the rank-preserving dissimilarity scenario,
n
the request rates Fm
are drawn from a Zipf distribution
with the same exponent s for all nodes. The object
preference rank for first node is [1, 2..., M ] and is shifted
to the right by k(n−1) positions for node n, n = 1, . . . , N ,
where k ∈ [0, M − 1] is the shift parameter. For example,
the most preferable object for node n is the one with
index u = mod(k(n − 1), M ) and its object preference
rank is [u, u+1, ..., M, 1, ...u−1]. Table 2(b) lists the values
of tightness for various values of k. Notably, tightness
is a monotonically decreasing function of the shift parameter k for k values satisfying (N − 1)k + C < M .
This inequality is satisfied throughout our performance
evaluation of the placement strategies. As k increases in
this interval, the divergence in the content preferences
between any two nodes increases. Likewise, tightness is
monotonically decreasing in s; only the manipulation of
s yields larger changes in the tightness values. The two
parameters allow high flexibility in tuning the tightness
value, whereby s provides for larger change steps and k
the finer control of the value.
The two dissimilarity patterns reflect groups with very
different breadth in their members’ preferences. The
rank-preserving dissimilarity pattern points to groups
with quite similar and focused interests. This could be
the result of a membership in some thematic-oriented
online association. It could also be due to trusting the
same popular websites for getting informed about content of interest (e.g., music or sport). In social networks
and blogs most popular users tend to influence the
majority of passive users, effectively amortizing variations in interests and preferences [6]. On the other hand,
shape-preserving similarity loosely points to groups,
whose members’ interests are spread over a wider range
of content, with the relative intensities being about
the same. For example, someone who likes listening
to music in her leisure time might download musicrelated content with the same normalized intensity as
6
Table 2: Example tightness values when M = 50 and N = 5
(a) rank-preserving
p
0.0
0.05
0.1
0.2
0.4
0.6
0.8
1.0
T
∞
45.74
10.17
2.09
0.46
0.24
0.17
0.14
k
0
1
2
3
4
5
6
7
(b) shape-preserving
s=1
s=0.5
s=0.1
T
k
T
k
T
∞
0
∞
0
∞
0.37
1
2.24
1
83.42
0.27
2
1.49
2
52.09
0.23
3
1.22
3
40.93
0.21
4
1.07
4
35.15
0.20
5
0.98
5
31.65
0.19
6
0.93
6
29.38
0.18
7
0.89
7
27.86
someone interested in sports would download sportsrelated content.
Nevertheless, these two patterns have been chosen
primarily because they let us control the level of preferences’ dissimilarity in our experimentation rather than
thanks to their representative power. The parameters
p, for the rank-preserving dissimilarity, and {k,s}, for
the shape-preserving dissimilarity, serve as tuning knobs
with predictable effect. Tuning these parameters, we
can synthesize a broad range of possible dissimilarity
patterns across the social group. Last but not least, these
special similarity patterns let us gain further insights
as to how well the self-aware cooperative placement
strategy approximates the optimal one in special cases.
We exercise this flexibility in the analysis that follows
in Section 5 and the numerical evaluation in Section
6; later, in Section 7, we experiment with real-world
content preference profiles drawn from an online social
bookmarking application.
5
P LACEMENT COST UNDER THE SELFISH
AND SELF - AWARE COOPERATIVE STRATEGIES
The aggregate content access cost under the altruistic
strategy can be numerically computed, at least for small
values of M, N , by solving the ILP problem (2.4) in
Section 2. Herein, we derive analytical expressions for
the per-node access cost under the selfish and self-aware
cooperative content placement strategies for the two
content preference-dissimilarity patterns described in
Section 4. Apparently, the costs under the altruistic and
selfish strategies constitute lower and upper bounds,
respectively, for the overall cost of the self-aware cooperative strategy.
In our analysis, the group nodes are indexed in order
of increasing Zipf distribution exponent s (for the rankpreserving dissimilarity) and distribution-shift k (for the
shape-preserving dissimilarity). For content items, on the
other hand, two kinds of item indexing become relevant:
the ”global” one, enumerating all objects in decreasing
preference order of node 1; and, the local node-specific
ones, which index objects in decreasing preference order
of the respective nodes. The two types of indexing
coincide under rank-preserving dissimilarity; whereas
there are N different local indexings, one per node,
under shape-preserving dissimilarity. The indexing type
of relevance in each case should be apparent from the
context.
5.1 Selfish placements
When the nodes behave selfishly, it is possible to analytically compute the amount of content they access
from the three levels of data storage, i.e., locally Cl ,
remotely from the caches of the group member nodes
Cr , and externally from server(s) or nodes in other social
groups Cs , as well as the resulting access costs. We
treat separately the two generic dissimilarity patterns
described in Section 4.
Rank-preserving dissimilarity, p 6= 0: All nodes store
locally the same C content items that commonly rank
top at their preferences and access the remaining M − C
items from external sources. What changes with p is the
preference amount that is concentrated in the C items
each node stores, which is controlled by the exponent
sn = p·(n−1) in the Zipf content preference distribution.
Therefore, Cl = C, Cr = 0, and Cs = M − C for all nodes
and the overall access cost for node n, is given by
PM
PC −sn
−sn
j=C+1 j
j=1 j
S
tl + P M
ts
(5.1.1)
Cn (P) = PM
−sn
−sn
j=1 j
j=1 j
PM
PC −p·(n−1)
−p·(n−1)
j=C+1 j
j=1 j
tl + P M
ts .
= PM
−p·(n−1)
−p·(n−1)
j=1 j
j=1 j
Shape-preserving dissimilarity, k 6= 0: Now, the C items
each node stores locally are, generally, different than
those other nodes store. Assuming that the number
of content items exceeds the cumulative group storage
capacity, M ≫ N ·C, we can distinguish two possibilities:
a) k < C: There is partial overlapping in the preferences of two (or more) nodes (Fig. 1(a)). Each node
accesses Cr = (N − 1)k items from the storage of the
other group nodes and Cs = M − C − (N − 1)k items
from the server(s). As the shift parameter k increases,
the nodes of the social group cumulatively store and
can access from each others’ storage more content. Yet,
nodes with higher index n have to access more items
ranking higher at their preferences from external sources
since they are not stored by any other group member.
Worst of all, the node N has to fetch the content items
ranking at positions [(C + 1), (M − (N − 1)k)] in its own
preference distribution at cost ts , whereas he can get
access to content objects in positions [M −(N −1)k+1, M ]
through the storage of the other group nodes.
The content access cost for node n can be written
PM−(n−1)k
PC −s
−s
j=C+(N −n)k+1 j
j=1 j
S
tl +
ts (5.1.2)
Cn (P) = PM
PM −s
−s
j=1 j
j=1 j
PM
PC+(N −n)k −s
−s j
j=M−(n−1)k+1 j
j=C+1
tr .
+
+
PM −s
PM −s
j=1 j
j=1 j
b) k > C: There is no overlapping in the items each
node stores locally in its cache (Fig. 1(b)) so that N · C
different content items are stored within the group. The
rest of the Cs = M −N ·C objects have to be fetched from
external sources. As long as N · k + C ≤ M , the access
cost increases with higher k and node index n values.
7
1 2 3 4 5 6 7 8 9 10 .. ..
M
…
1 2 3 4 5 6 7 8 9 10 .. ..
index of objects
C
…
node 1
node 1
k
k
…
node 2
node 2
M
index of objects
C
access from other
nodes’ storage
access from
local storage
access from other
access from server
nodes’ storage
access from other
nodes’ storage
…
access from
local storage
…
node n
node n
access from server
…
…
node N
node N
(a) k < C, partial overlapping in local storage
(b) k ≥ C, each nodes stores different items
Figure 1: Objects stored at different group nodes under shape-preserving dissimilarity.
PC
j=1
j −s
j=1
j −s
CnS (P) = PM
tl +
P
PM −i(k+C)+1
N−n
n−1
X i(C+k)+C
i−s X
j −s
j=i(C+k)+1
j=M −ik−(i−1)C+1
+
tr
PM −s
PM −s
i=1
j=1
j
j=1
i=1
j
Pi(C+k)
PM −ik−(i−1)C
PM −(n−1)(C+k) −s N−n
n−1
X j=iC+(i−1)k+1
j −s X
j −s
j
j=M −(i−1)(k+C)
(N−n+1)(C+k)+1
+
+
+
ts .
PM −s
PM −s
PM −s
i=1
j=1
j
i=1
The resulting content access cost for node n is given by
(5.1.3).
Identical uniform content preferences: This represents an
extreme case of zero dissimilarity in the preferences of
group nodes. It results from the general rank-preserving
dissimilarity scenarios when p = 0, and from the shapepreserving dissimilarity scenarios, when s = 0.
Contrary to the general dissimilarity scenarios, the
distribution of content objects at the three storage levels
and the corresponding access cost are no longer deterministic. The C objects stored locally at each node are
randomly chosen out of the full set of M objects so that
the total number of different objects collectively stored at
the caches of the N group nodes is a random variable X,
C ≤ X ≤ M . Each node now accesses C objects from its
own cache, X −C objects from the caches of the other N 1 group nodes and the remaining M − X content objects
from the server and the expected per-node content access
cost is given by
CnS (P) =
1
[C ·tl +(E[X]−C)·tr +(M −E[X])·ts ] (5.1.4)
M
Whereas the probability distribution of X is more
involved and given in [14], to compute the expected
value E[X], it suffices to remark that the selection or not
of each object by a single node
isMa Bernoulli trial with
success probability ps = M−1
C−1 / C = C/M . Therefore,
the selection of each object by at least one out of the N
nodes equals 1 − (1 − ps )N and
E[X] = M (1 − (1 − C/M )N )
(5.1.5)
j=1
j
j=1
j
(5.1.3)
5.2 Self-aware cooperative placements
The starting point for the self-aware cooperative placements are the selfish placements of the first step. In the
second step of the strategy, nodes take turn in adjusting
the selfish placements of the first step evicting objects
that are replicated elsewhere in the group and inserting
new ones, not yet stored anywhere in the group, inline
with the properties (P1)-(P3) listed in Section 2.1 and
proven in [19]. In general, by the end of the second step,
each node n has retained the first jn objects of the initial
selfish placement and inserted C − jn new ones, which
are not replicated anywhere else in the group, according
to property (P2).
5.2.1 Rank-preserving dissimilarity
By the end of the first step, the selfish strategy gives rise
to full replication of the same C objects in the group
so that candidates for insertion are objects with global
indices in [C + 1, M ]. Let si be the exponent in the Zipf
preference distribution for node i and consider the first
node just before executing the second step (placement
adjustment). Depending on its distribution skewness,
node 1 will retain j1 objects in its cache, 1 ≤ j1 ≤ C,
and remove the rest, where
j1
ts − tl
tr − tl
>
}, C)
us1
(2C − u + 1)s1
2C + 1
⌋, C)
(5.2.1)
= min(⌊
l 1/s1
1 + ( ttrs −t
−tl )
= min(max{u :
namely, j1 equals the maximum item index of those
stored locally in the first step, for which the benefit of
retaining it in the local storage exceeds the insertion
benefit from the next most preferred item among those
not yet stored elsewhere in the group (Fig. 2).
8
Since nodes only insert non-represented objects in this
step (P2), candidates for insertion by the second node
will be the objects [2C − j1 + 1, 3C −
1 ], and generalizing,
Pjn−1
by the nth node, the objects [nC − l=1
jl + 1, (n + 1)C −
Pn−1
j
].
The
number
of
items
retained
locally by node
l=1 l
n is
jn
tr − tl
usn
ts − tl
>
}, C)
Pn−1
[(n + 1)C − i=1
ji − u + 1]sn
Pn−1
(n + 1)C − i=1
ji + 1
⌋, C) (5.2.2)
= min(⌊
l 1/sn
)
1 + ( ttrs −t
−tl
= min(max{u :
Moreover, the index of the last inserted object in the
group, after all N nodes have finished with their second
P −1
step, is inmax = N C − N
i=1 ji . If we denote jmin =
mini {ji }, we can state the following property for the
structure of the content placement at the caches of the
group nodes.
Property 5.1. Under the self-aware cooperative placement
strategy and rank-preserving dissimilarity across nodes’ preferences, the number of replicas within the group is: N for
objects with (global) indices in [1, jmin ] (full replication);
r, 1 ≤ r < N , for objects with indices in [jmin + 1, C]; one
for objects with indices in [C + 1, inmax]; and, zero for objects
with indices in [inmax + 1, M ].
Drawing on this property, we can derive conditions
for two particular instances of the placements.
Proposition 5.1. The original selfish placements of all nodes
remain intact during the second step when
ts − tl 1/smin
< 1 + 1/C
)
(
tr − tl
where smin = minj {sj }.
(5.2.3)
Likewise, we can derive the condition for inserting
(N −1)C items during the second step so that eventually
N C different items be stored within the group.
Proposition 5.2. Under the self-aware cooperative placement
strategy and rank-preserving dissimilarity, N · C different
content objects will be stored in the caches of the group nodes,
each represented only once, if
l
log( ttrs −t
−tl )
(5.2.5)
smax <
log(N C)
Proof: To have all objects only once replicated within
the group, nodes [1,N-1] should replace all C items of
their original selfish placements with non-represented
ones. Therefore, objects [nC +1, (n+1)C], 1 ≤ n ≤ (N −2)
should be inserted at node n, whereas node N will
always retain its original selfish placement since these
most preferred objects will not be replicated anywhere
else in the group(P1).
The worst-case setting is that the node playing in position N −1 is the node with the most skewed distribution,
smax . Therefore, for a placement with N C different items
in the group caches, irrespective of the order of play,
it suffices that the object N C replaces the first most
preferred object of node N − 1,
tr − tl
ts − tl
> smax .
(5.2.6)
s
max
(N C)
1
Solving for smax yields (5.2.5).
With these results at hand, we can compute the pernode access cost as
CnC (P)
=
jn
X
i=1
M
X
i−sn
tl +
i
−sn
i=1
in
max
X
M
X
i−sn
i=inmax
M
X
i
ts
(5.2.7)
−sn
i=1
Pn
j
X l=1 l
(n+1)C−
i−sn
i−sn −
Proof: As can be seen in (5.2.4) and (5.2.2), the
P
n−1
i=jn +1
i=nC−
jl +1
number of objects retained by node n is a monotonically
l=1
+
tr
M
increasing function of s. In the same time, it depends on
X
i−sn
its order of play, i.e., which nodes have preceded him in
i=1
adjusting their placements. For two nodes k and l, with
k playing before l and sk ≤ sl , (5.2.2) suggests that
whereby the objects that a node eventually accesses from
the other group nodes’ caches are the full set of nonPk−1
Pl−1
(k + 1)C − i=1 ji + 1
(l + 1)C − i=1 ji + 1
jk =
≥
= jl represented objects that are inserted at the second step
ts −tl 1/sl
l 1/sk
)
)
1 + ( ttrs −t
1
+
(
of the strategy(algorithm) minus those locally stored at
−tl
tr −tl
that node.
The self-aware cooperative placements will coincide
with the selfish ones only if jn = C, ∀n ∈ N . The nec- 5.2.2 Shape-preserving dissimilarity
essary and sufficient condition for this is that jmin = C, As with selfish placements, we need to distinguish beor equivalently that the node with the least skewed tween two cases:
distribution of preferences does not evict any item given
a) k > C: assuming that N (C + k) ≤ M , the selfish
that nodes playing before it have not done so either. placements in the first step result in the placement of N C
Hence, it must hold that
different objects in the caches of the group nodes, each
one represented only once. According to (P1), nodes do
2C + 1
⌋, C) = C
(5.2.4) not evict objects from their caches; hence, the placements
min(⌊
l 1/smin
1 + ( ttrs −t
)
−tl
under the self-aware cooperative placement coincide
which directly yields (5.2.3).
with those under the selfish strategy.
9
..
..
the last node should not (find it profitable to) evict
its least preferable object that is replicated in the
group.
The first condition translates to
•
i =1
inmax
3C-j1-j2
C
2C-j1
1 2 3 4 ..
C+1
n
( n + 1)C − ∑ ji
M
index of objects
node 1
j1
ts − tl
C +k+1
ts − tl 1/s
tr − tl
>
⇒
)
>(
Cs
(C + k + 1)s
C
tr − tl
node 2
j2
access from
local storage
node n
access from
external sources
whereas the second condition can be expressed as
tr − tl
ts − tl
C+1
ts − tl 1/s
>
⇒
)
>(
s
s
(C − k)
(C + 1)
C −k
tr − tl
jn
node N
access from other
nodes’ storage
Evicted content items
Figure 2: Self-aware cooperative strategy, rank-preserving dissimilarity: evicted and inserted content items per node.
Proposition 5.3. Under shape-preserving dissimilarity and
k > C the self-aware cooperative placement coincides with
the optimal.
Proof: For k > C, not only the self-aware cooperative
but also the optimal placement coincides with the selfish
one. Since the selfish placement comprises N C discrete
content objects, the optimal placement could, in principle, differentiate in three ways: a) replicate one of the
content objects more than once; b) mutually exchange
two content objects stored in different nodes; c) replace
one of the objects in the selfish placement with one of
those not represented in the placement, i.e., objects with
indices in [nC + 1, (n + 1)C − k]. It is straightforward
to see that the global utility generated in each one of
three cases is lower than that accruing through the selfish
placement. Therefore, the two placements coincide.
b) k < C: the selfish placements give rise to overlaps in
the contents of nodes’ caches. The number of different
objects that are placed in the whole group are md =
(n− 1)k + C and their replication count varies in [1, ⌊ Ck ⌋].
Moreover, for k < ⌊C/2⌋, the md − 2k objects stored by
the group nodes feature at least two replicas and could
be evicted by one or more group nodes in the second
step.
Whereas, the exact computation of the per-node access
cost in this case is cumbersome, it is easier to prove the
following result.
Proposition 5.4. The placements under the self-aware cooperative strategy and shape-preserving dissimilarity across nodes’
preferences coincide with the selfish placements when
ts − tl 1/s
k/C > (
) − (1 + 1/C)
(5.2.8)
tr − tl
Proof: The two placements will coincide as long
as no object evictions/insertions are made during the
second step. Given that the local indices of objects
[md + 1, M ], hence their preference rank, increases for
higher node indices, two conditions should be met so
that nodes’ original placements do not change.
th
• the (N − 1)
node should not (find it profitable to)
evict its least preferable object currently in its cache.
(5.2.9)
since (C + 1)/(C − k) > (C + k + 1)/C, ∀k > 0, the
first inequality is the active constraint and (5.2.8) results
trivially.
Therefore, equations (5.2.3) and (5.2.8) already suggest
that the placements emerging under the self-aware cooperative and selfish strategies tend to coincide as the
exponents of the Zipf preference distributions (s1 ∝ p
for rank-preserving dissimilarity) and shift parameter k
increase. In other words, since tightness decreases with s
and k (see Table 2), the gain under cooperation fades out
as the content preferences of nodes diverge. We elaborate
on this result in the next section.
Identical uniform content preferences: In this extreme
case of demand distributions, the self-aware cooperative
strategy will generate a placement of N C different content objects, each one represented only once in the union
of the group’s caches.
Proposition 5.5. The cost of the self-aware cooperative placement under identical uniform content preferences equals the
optimal one.
Proof: Irrespective of the order of play, nodes evict
objects that are elsewhere represented in the group and
replace them with equally wanted objects that were not
selected by any group node in the first
of self round M!
QN −1
ish placements. There are i=0 M−iC
=
C!N (M−N C)!
C
different possible placements with the same social cost,
which coincides with the best possible.
6
R ESULTS
AND DISCUSSION
The numerical examples in this section illustrate how
group similarity, aka tightness, shapes the tradeoffs induced by the three behavior-based content placement
strategies. Therefore, they help establish guidelines as
to which behavior (strategy) would be beneficial to
individual nodes and/or the entire group, under given
similarity levels in the preferences of the nodes in the
social group.
In the numerical examples in this paper initially N = 5
nodes; a small number of nodes helps us better illustrate
and discuss the results regarding the content access
cost for each node. The default value for node storage
capacity is C = 10 objects, for object population M = 50
objects and for the costs tl = 0, tr = 10 and ts = 20 cost
units.
10
T=∞
Selfish
Self-aware cooperative
Optimally altruistic
20
15
Access cost
Access cost
20
for different values of tightness, T . We discuss these two
viewpoints for very high and very low tightness values.
T=∞
Selfish
Self-aware cooperative
Optimally altruistic
10
5
15
10
5
0
0
1
2
3
4
5
1
2
Node id
3
4
5
Node id
(a)
T=2.09
20
Selfish
Self-aware cooperative
Optimally altruistic
20
15
Access cost
Access cost
T=0.27
Selfish
Self-aware cooperative
Optimally altruistic
10
5
15
10
5
0
0
1
2
3
Node id
4
5
1
2
3
Node id
4
5
(b)
T=0.24
Selfish
Self-aware cooperative
Optimally altruistic
20
15
Access cost
Access cost
T=0.21
Selfish
Self-aware cooperative
Optimally altruistic
20
10
5
15
10
5
0
0
1
2
3
4
5
1
2
Node id
3
4
5
Node id
(c)
T=0.14
Selfish
Self-aware cooperative
Optimally altruistic
20
Access cost
15
10
5
15
10
5
0
0
1
2
3
Node id
4
5
1
2
3
Node id
4
5
(d)
Figure 3: Individual access cost under the three content placement strategies for different values of tightness T , under rankpreserving (figures on the left) and shape-preserving (figures
on the right) dissimilarity.
Selfish
Self-aware cooperative
Optimally altruistic
70
60
50
40
30
20
10
80
70
Total access cost
80
Total access cost
Access cost
T=0.19
Selfish
Self-aware cooperative
Optimally altruistic
20
60
Selfish
Self-aware cooperative
Optimally altruistic
50
40
30
20
10
0
0
∞
2.09 0.46 0.24 0.17 0.14
T
(a) Rank-preserving
∞ 0.37 0.27 0.23 0.21 0.20 0.19 ≃ 0
T
(b) Shape-preserving
Figure 4: Total access cost vs. tightness T.
6.1 Placement strategy comparison
Figures 3 and 4 consider separately the two scenarios
for preferences’ dissimilarity in Section 4. They plot the
per-node and aggregate (i.e., for the entire group) access
cost under the three content placement strategies and
6.1.1 Social groups with infinite or very high tightness
The first important remark out of the two plots is that
the optimally altruistic strategy outperforms the other
two regarding not only the cost for the entire group
(by definition), but also the cost for individual group
nodes (Fig. 3(a)). This relation holds irrespectively of the
dissimilarity scenario in question and suggests that the
optimally altruistic behavior is the clear winner-behavior
for every node in a very tight social group.
The second noteworthy outcome is that the access
costs for both individual nodes and the entire group
under the self-aware cooperative strategy are very close
to (slightly higher than) those under the optimally altruistic strategy. Since its implementation is significantly
simpler than the optimal altruistic one, it emerges as
the favorite alternative for node-members of tight social
groups, achieving a good tradeoff between performance
and complexity.
Looking closer into the rank-preserving dissimilarity
plots, when tightness approaches infinity (Fig. 3(a) on
the left), the access cost for all group nodes under the
self-aware cooperative and optimally altruistic strategies
tend to become equal. Infinite tightness in the general
case (p = 0, s 6= 0 in Section 4) implies that a given
object is requested with the same intensity by all nodes.
Both the self-aware cooperative (in line with Proposition
2) and the optimally altruistic strategies end up with
the same placements P, which insert many items and
spread them across the storage capacity of the group
nodes; contrary to the selfish placements, which blindly
replicate the same C objects N times.
Summarizing, the tighter the social group, the more
incentives the nodes have to behave in cooperative or,
even, altruistic rather than selfish manner.
6.1.2 Social groups with low tightness
The first conclusion out of Fig. 3 concerns the way
the total access cost is spread across the group nodes,
regardless of the placement strategy, under the two types
of dissimilarity and for low T values. The content access
cost split under rank-preserving dissimilarity is uneven.
Nodes with higher indices “pay” much less than nodes
with smaller indices, irrespective of the adopted content
placement strategy. This unfair cost distribution becomes
more pronounced as tightness decreases. The reason
behind this unfairness has to do with the demand distributions of the five nodes. Remember from Section 4 that
the higher the node index the more skewed the node
preference distribution. Hence, the preference of nodes is
more concentrated around fewer top-ranked objects and
there is less demand for the remaining objects that have
to be accessed from remote storage, whether from the
N − 1 group nodes at cost tr or outside the group at cost
ts . On the contrary, under the shape-preserving dissimilarity scenarios, the total access cost is more uniformly split
11
6.2 Responsiveness of the self-aware cooperative
strategy to changes in user preferences
In this set of simulation experiments we study how fast
the self-aware cooperative strategy responds to changes
in the content preferences of users. Initially the nodes’
caches are empty. Nodes learn the content preferences of
users over time and every time they receive a number
of requests, say R, they run one iteration of the selfaware cooperative strategy; namely, each node n updates
its cache with the Cn most requested content items
(selfish first step) and then takes turn into adjusting its
placement through evictions and insertions of new items,
based on its running estimates for the user’s interest in
them. We let N = 10, M = 10000 items and Cn = C = 10
for all nodes.
In the course of the simulation, we change the content
preference distributions of users twice. In the beginning, all nodes receive requests for content following
the Zipf distribution with exponent s = 1, i.e., there
220
Simulation experiment
Total access cost
200
rank−reversing
shift−prerefence
180
160
140
Expected total
access cost
r(a)
120
100
0
a%
5000 10000 15000 20000 25000 30000 35000 40000
Number of received requests
Figure 5: Performance of self-aware cooperative as function of
number of received requests.
5000
4500
4000
rank−reversing
shift−preference
3500
r(6%)
across the group nodes and smaller in absolute values.
There is far more diversity in the content that different
nodes are interested in.
Contrary to high tightness scenarios, Fig. 3 and 4 suggest that neither cooperation nor altruism are attractive
options when the node preferences are highly diverse.
Firstly, the performance of the self-aware cooperative
and altruistic strategies approaches that of the selfish one
as tightness decreases. Nodes’ interests are focused on
fewer objects and, thus, they do not gain much by adjusting their placements through eviction and insertion
of other items.
Secondly, the altruistic strategy cannot avoid mistreatment of individual nodes; for example, this is the case
with Node 5 in Fig. 3(d), for both preference dissimilarity patterns we consider. Two are the straightforward
remarks when looking closer at Fig. 3: a) the number
of mistreated nodes increases as the value of T drops;
b) it is mainly the nodes with higher indices that are
mistreated (this is more apparent for the rank-preserving
dissimilarity scenarios). The optimally altruistic strategy
shuffles the objects in a more radical way so that some
nodes may end up replacing their C top-ranked items
and raise significantly their own access cost in favor of
the aggregate access cost minimization.
Note that in the previous results, the nodes play in
specific order. Whereas the order of play affects significantly the access cost experienced by individual nodes
[14] [19], it has a much milder impact on the aggregate
access cost. Table 3 reports the minimum and maximum
values for the total access cost over all 120 possible
permutations in the order the five nodes update their
caches in the second round of the self-aware cooperative
strategy. The variance of this cost with the order of
play is negligible compared to the change with the
Zipf distribution parameter and the shift parameter k.
Relevant results can be found in [14].
3000
2500
2000
1500
1000
0
100 200
400 500
1.000
R
Figure 6: Access cost convergence time vs.period of placement
algorithm execution, measured in number of requests.
is high similarity in the content preference across the
group nodes and the cooperation benefits are maximized. After 2000 requests are received by all nodes,
we reverse the content preference distributions so that
n
n
F ′ m = FM−m+1
, ∀m ∈ M and n ∈ N (rank-reversing).
In Fig. 5, we see that temporarily, till nodes gradually
inform their caches and converge to the right placements, the nodes spend much more than the average
expected cost for accessing content. Each plotted point
is the cost averaged over the last 50 requests received by
all nodes and 100 simulation runs. The system gradually
improves its performance and drops the running average
access cost to within a = 6% of the expected access
cost (113.93) after r(a) = 1719 requests. Note that the
total expected access cost remains the same in this case
since the distributions are symmetric. After further 18000
requests, the preference distributions change again, this
time following the shape-preserving dissimilarity pattern with shift value k = C/2 (shift-preference). This time
the overshoot of the running aggregate access cost is
milder than the first change; yet it takes the algorithm
4232 requests before the cost converges to a = 6% of the
steady-state expected aggregate access cost.
The frequency of placement adjustments (i.e., iterations of the self-aware cooperative algorithm) induces
a tradeoff between the excess access cost in the transient
phase, during which the nodes fill their caches in response to their estimates about the nodes’ preferences,
and the overhead of the algorithm execution, both in
terms of computations and network resources.
12
Table 3: Maximum and minimum values of total access cost over the 120 permutations in the order nodes update their caches
in the 2nd round of the the self-aware cooperative strategy: variable T , M = 50, and N = 5.
Tightness (T)
∞
45.74
10.17
2.09
0.46
0.24
0.17
(a) rank-preserving
Max total access cost
Min total access cost
40.0000
40.0000
40.4886
39.4035
42.8774
39.2597
42.9292
40.1894
34.8047
33.1579
27.4510
26.1728
22.8114
22.3532
Figure 6 plots the access cost convergence time r(6%)
against the period of algorithm execution R, both measured in number of requests, for the rank-reversing and
shift-preference types of content preference changes. Although frequent invocations of the algorithm can accelerate the convergence of the algorithm upon disruptive
changes in the users’ preferences, the performance gap
between lower and higher R values is rather moderate.
Considering that these scenarios represent extreme cases
of content preference changes across the social group
population, we conclude that the self-aware cooperative
strategy can respond rather fast to stochastic changes of
users’ content preferences even at moderate execution
frequencies.
7 A PPLICATION TO A REAL NETWORK
We apply the selfish and the self-aware cooperative strategy to data traces extracted from the Delicious website
(www.delicious.com). Delicious is a social bookmarking
application where users can save all their web bookmarks (annotated with tags) online, share them with
other users, and track what other users are bookmarking
themselves. Each Delicious user together with other
users, who have subscribed to see her/his bookmarked
web pages, effectively forms a network. We draw on the
organization of users into networks and their interests
into tags to generate user interest distributions and set
the content item population size, M , equal to the number
of different tags in each user-network. The purpose of
this example is to assess the relation between interest
similarity of user groups and achievable cooperation
gains under ”real world” interest dissimilarity patterns
beyond the synthetic ones we introduced in Section 4
and used as references for our analysis and experimentation in Sections 5 and 6.
From user interest profiles to interest distributions.
Let M be the set of most popular tags used by each
n
Delicious user. Let Bm
be the number of bookmarks
tagged with m (1 ≤ m ≤ M ) by user n (1 ≤ n ≤ N ).
Then the (normalized) interest of node n in tag m is
given by the ratio of the number of bookmarks tagged
with m by node n over the total number of bookmarks
of this user:
Bn
n
.
(7.1)
Fm
= PM m
n
m=1 Bm
Experimentation set-up. The Delicious network is
crawled in two ways. As we shall see, different ways
Tightness (T)
∞
0.37
0.27
0.23
0.21
0.20
0.19
(b) shape-preserving
Max total access cost
Min total access cost
29.2582
29.2582
29.2925
28.2860
28.7365
27.6922
27.1734
27.1091
26.1813
25.6366
24.6086
24.6086
23.5663
23.5663
of crawling the network can derive user groups with
different tightnesses. The first method starts from four
Delicious accounts (root users) chosen randomly from
the website. From each root user 29 users, who follow the
root user, are extracted using a breadth-first exploration
of the graph formed by these links. We consider that
each of these users has a capacity of 10 bookmarks in
their cache. To avoid the long tail of infrequently used
tags, only bookmarks that contain the 99 most popular
tags (or objects) are considered for each user. The interest
profiles of 120 in total users are derived from (7.1). The
tightness of this group of nodes is 0.0956, which shows
that common interests are not the primary reason why
nodes choose to follow other nodes. Running the selfish
and self-aware cooperative strategy for this group results
in gain under cooperation 1.0459. This gain is similar
to that obtained for low similarity under the rank- and
shape-preserving dissimilarity patterns in [14], Fig. 1 and
2.
The second procedure is similar to the first one; only
now the four root users are selected among those having
placed recent bookmarks on the website and we retain the
30 highest preference tags. The tightness and gain under
cooperation for this case are computed to be 0.1420 and
1.5240, respectively. As expected, they are higher since
many users are interested in the same tags.
Overall, the relation between interest similarity and
cooperation gain, as analyzed in Section 6, pertains also
under the dissimilarity patterns that emerge from the
Delicious user-networks. On a secondary note, the tightness values of these sample networks are low, implying
that they do not avail strong interest similarity structure.
This is a subject worth investigating further, our first
results being reported in [13].
8
R ELATED
WORK
Algorithms for file sharing between computers have
mostly been studied in the context of data replication
[21]. Data replication refers to the storage of files or, more
generally, information objects, in specific points in a network, so that they can be retrieved by requested nodes
at smaller access costs. Earlier research in this area has
mostly considered centralized implementations [7], [20]
of file placements. However, in modern networks (e.g.,
ad-hoc, p2p, opportunistic networks) as both the number
of nodes increases and they become more autonomous,
13
distributed algorithms become more relevant and important [19], [32]. The distributed selfish replication game
is introduced and studied in [19], where the authors
propose an algorithm for its solution and analyze its
main properties. In [19] the assumption is that all nodes
within a group can communicate and cooperate with
each other. More recently, Pacifici and Dan in [25] relax
this assumption and consider replication games over
arbitrary social graphs, which capture possible topological constraints on the possible interaction between the
players. They derive sufficient conditions for letting the
players reach an equilibrium of the game and propose
a distributed algorithm in this respect. On the other
hand, Borst et al. in [4] assume altruistic players making
placements that maximize the aggregate benefit over the
whole network rather than theirs. The performance of
their greedy algorithm is within a constant factor of two
from the globally optimal performance under arbitrary
demands and, even closer, within 1.33 of the optimal
under identical content preferences and uniform cache
capacities.
The self-aware cooperative strategy we consider coincides with the distributed algorithm in [19]. The group
nodes are rationally selfish and perfectly connected and
seek to maximize their own benefit from the objects they
select to store in their caches. Contrary to prior work, our
focus is set on the behavioral aspects of the algorithms,
i.e., selfish, cooperative, altruistic. We introduce a measure, tightness, which captures the degree of similarity
in the preferences of nodes within a social group, and
assess its impact on the relative performance of these
algorithms(behaviors) and the resulting tradeoffs among
them.
Data replication has also been studied in the context
of mobile social networks, with social characteristics
being embedded into data replication algorithms. In
[3], the authors construct a dynamic learning algorithm
where nodes from various social communities opt for
a utility-maximizing content placement strategy based
on their encounters with other nodes. The content utility is related to the availability of content in different
communities, as well as the ties a user has with each
community. In [5] the authors study how content is
distributed in an opportunistic network considering both
technical constraints (e.g., battery/proccesing power and
wireless bandwidth) and user preferences. In [15] the
authors propose an approach that can enhance content
dissemination by associating both interest- and localitybased dynamics of social groups. Finally, one of the main
contributions of [18] is the development of a model for
assessing the impact of users’ characteristics (e.g., interest
in content and willingness to share it) on the potential
gains achievable through opportunistic contacts.
called selfish, self-aware cooperative and optimally altruistic, respectively. As their names suggest, these strategies reflect three fundamental behavioral paradigms in
networked communications.
Our results suggest that the level of similarity in
nodes’ preferences across a social group is key to deciding which content placement strategy (i.e., what kind
of behavior) to follow. Altruism emerges as a winwin behavior only in tight social groups as long as
the implementation cost is not an issue: it minimizes
the content access cost not only collectively for the
whole group (by definition) but also for each individual
node. As tightness decreases, the collective group gain
under the altruistic and self-aware cooperative strategies
fades out, while certain nodes may be mistreated when
behaving altruistically. Therefore, and considering also
its low complexity, the selfish strategy becomes more
attractive. In summary, our evaluation shows that the
benefits of cooperation increase with the group tightness.
Therefore, an on a more practical note, tightness should
be used as a decision criterion: a) when choosing content
placement strategies under given group membership; or,
more broadly, b) for carrying out performance-driven
group management operations such as group formation/merging/splitting.
As a final note, it is worth mentioning that the positive
correlation between similarity and cooperation/altruism
is reported in studies of human social behavior [9].
Research results in literature suggest that cooperation
between individuals with similar characteristics evolves
over time to a stable strategy (behaviour) [1]. Further,
among the explanations evolutionary theorists have provided about the emergence of altruism in human behavior is reciprocal altruism, where individuals obtain
mutual benefits through their exchanges [8], [29]. In our
application, the similarity in the interests of nodes (i.e.,
high tightness) could be seen as a catalyst for reciprocal altruism since it increases the chances of mutually
beneficial interactions.
10 ACKNOWLEDGMENTS
We thank Walter Colombo and Martin Chorley from
Cardiff University for kindly providing us with traces
they collected from the Delicious network. This work
has been supported by the European Commission ISTFET project SOCIALNETS (FP7-IST-217141) and the
the Marie Curie grant RETUNE (FP7-PEOPLE-2009-IEF255409).
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Eva Jaho is currently working as an IT consultant at Athens Technology
Center, Greece (www.atc.gr). She received the PhD and MsC in Networking from the Department of Informatics and Telecommunications
of the National & Kapodistrian University of Athens, Greece, in 2011
and 2007 respectively. She received the Diploma degree from the same
university department in 2005. She has participated in several European
research projects in the field of networking and telecommunications. Her
main research interests lie in the analysis of content networks and data
dissemination, as well as social networking applications.
Merkouris Karaliopoulos holds a Marie Curie Fellowship with the Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece. He was awarded his diploma in
Electrical and Computer Engineering from Aristotelian University of
Thessaloniki, Greece, in 1998, and his PhD in the same field from
the University of Surrey, UK, in 2004. He spent one year (2006) as
postdoctoral researcher in the Computer Science Dept. of University of
North Carolina at Chapel Hill, NC, USA and three years (2007-2010)
as Senior Researcher and Lecturer in the Swiss Federal Institute of
Technology (ETHZ), Zurich, Switzerland. His research interests lie in
the general area of wireless networking, currently focusing on aspects
of network and service resilience to imperfect node cooperation.
Prof. Ioannis Stavrakakis received the Diploma in Electrical Engineering from the Aristotelian University of Thessaloniki, Greece (1983), and
the Ph.D. in EE from the University of Virginia (1988). He was Assist.
Professor of CSEE, Univ. of Vermont, during 1988-1994, Assoc. Professor of ECE, Northeastern Univ., Boston, during 1994-1999, Assoc.
Professor of Informatics and Telecommunications, University of Athens,
Greece, during 1999-2002, and Professor since 2002. His research
interests are on resource allocation protocols and traffic management
for communication networks, with recent emphasis on: peer-to-peer,
mobile, ad hoc, autonomic, delay tolerant and future Internet networking.
His past research has been published in more than 180 scientific
journals and conference proceedings. He has organized several conferences, has been a Chairman of IFIP WG6.3, Assoc. Editor for the
IEEE/ACM Trans. on Networking, ACM/Springer Wireless Networks,
Computer Networks, and currently of Computer Communications Journals. He is currently Assoc. Dept Chair and Director of Graduate Studies
and an IEEE Fellow.